12.6 Correlation (Aggregating the CoV)

Overall

Assumes the 3 main risk components are independent of each other

\[\begin{equation} \phi = \sqrt{\phi_{indep}^2 + \phi_{internal}^2 + \phi_{external}^2} \tag{12.2} \end{equation}\]

Caveat of quantitative method to measure correlation:

  • Complexity > benefit

  • Results heavily influenced by past correlations

    (While future correlation may differ)

  • Difficult to separate past episodes of independent risk and systemic risk

  • Internal systemic risk cannot be modeled using standard correlation modeling techniques

  • Likely won’t aligned with our definitions of independent, internal/external systemic risks

Practical guidance:

  • Bucket into 0%, 25%, 50%, 75%, 100%

    (Any finer will likely lead to spurious accuracy)

  • Introduce dummy variables and see their impact in each risk/valuation group pair

    (Inflation, unemployment, propensity to suit, freq of CAT, fraud)

12.6.1 Independent Risk Cov Correlation

Assume independence across liabilities, where \(i\) is the different valuation groups

\[\begin{equation} \phi_{indep}^2 = \sum_i (\phi_i w_i)^2 = (\vec{\phi w})(\vec{\phi w})^T \tag{12.3} \end{equation}\]
  • \(w_i\) can be just claims liabilities or total liabilities (including premium liabilities)

12.6.2 Internal Systemic Risk Correlation

Focus on correlation within internal systemic risk

  • Can have correlation between the outstanding claim and premium liabilities for the same valuation group
\[\begin{equation} \phi_{internal}^2 = (\vec{\phi w}) \times \mathbf{\Sigma} \times (\vec{\phi w})^T \tag{12.4} \end{equation}\]

  • \(\mathbf{\Sigma}\) is the correlation matrix

  • Again the \(\vec{w}\) is the % of total liabilities

12.6.3 External Systemic Risk Correlation

We measure the CoV for each risk category for each valuation group

  • e.g. high inflation will be correlated across all valuation groups that have long tail or event risk across LoB

\(\mathbf{\Sigma}_c\) is the correlation matrix between valuation groups for each risk category \(c\)

For a given risk category \(c\), the CoV is:

\[\begin{equation} \phi_{external, c}^2 = (\vec{\phi_{c} w}) \times \mathbf{\Sigma}_c \times (\vec{\phi_{c} w})^T \tag{12.5} \end{equation}\]

Then assume independence between risk categories:

\[\begin{equation} \phi_{external}^2 = \sum \limits_{c \: \in risk \: category} \phi_{external, c}^2 \tag{12.6} \end{equation}\]
  • Important to pick risk categories that are likely to be independent of each other